Arkiv för Matematik

Volume 61 (2023)

Number 1

Removability of product sets for Sobolev functions in the plane

Pages: 67 – 80

DOI: https://dx.doi.org/10.4310/ARKIV.2023.v61.n1.a4

Authors

Ugo Bindini (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

Tapio Rajala (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

Abstract

We study conditions on closed sets $C, F \subset \mathbb{R}$ making the product $C \times F$ removable or non-removable for $W^{1,p}$. The main results show that the Hausdorff-dimension of the smaller dimensional component $C$ determines a critical exponent above which the product is removable for some positive measure sets $F$, but below which the product is not removable for another collection of positive measure totally disconnected sets $F$. Moreover, if the set $C$ is Ahlfors-regular, the above removability holds for any totally disconnected $F$.

2010 Mathematics Subject Classification

46E35

Full Text (PDF format)

The authors acknowledge the support from the Academy of Finland, grant no. 314789.

Received 18 November 2021

Received revised 5 June 2022

Accepted 13 July 2022

Published 26 April 2023